Stability of the boundary-value problem for harmonic mappings

Geodesics in the space of relatively Kähler metrics

“…This will be done by linearising the WZW equation at a solution, and showing the derivative is an isomorphism. This has been carried out by Goldstein [25] for the standard harmonic map equation; adapting his proof, we show: Proposition 6.8. For 𝑟 ⩾ 2 and 𝛼 ∈ [0, 1], let 𝐹 ∶ 𝐶 𝑟,𝛼 (𝐷, 𝐺∕𝐾) → 𝐶 𝑟−2,𝛼 (𝐷, 𝑇(𝐺∕𝐾)) × 𝐶 𝑟,𝛼 (𝜕𝐷, 𝐺∕𝐾) be given by 𝐹(𝑓) = ( τ(𝑓), 𝑓| 𝜕𝐷 ), where τ(𝑓) ∶= tr(∇𝑑𝑓) + [𝐷 𝑢 𝑓, 𝐷 𝑣 𝑓] is the WZW operator (20).…”

“…This will be done by linearising the WZW equation at a solution, and showing the derivative is an isomorphism. This has been carried out by Goldstein [25] for the standard harmonic map equation; adapting his proof, we show: Proposition For $$r\geqslant 2$$ and $$\alpha \in [0,1]$$, let $$$$\begin{equation*} F:C^{r,\alpha }(D,G/K)\rightarrow C^{r-2,\alpha }(D,T(G/K))\times C^{r,\alpha }(\partial D,G/K) \end{equation*}$$$$be given by $$$$\begin{equation*} F(f)=(\tilde{\tau }(f),f|_{\partial D}), \end{equation*}$$$$where $$\tilde{\tau }(f):=\mathrm{tr}(\nabla df)+[D_uf,D_vf]$$ is the WZW operator (). If $$f:D\rightarrow G/K$$ is a solution to the WZW equation, then $$dF|_f$$ is an isomorphism.…”

“…Proof Following [25], we show the derivative is both injective and surjective at a solution. In fact, surjectivity follows from abstract index theory, given that the leading order term in the linearisation of the WZW equation agrees with that of the harmonic map equation; see [25, pp. 353–355].…”

“…In these coordinates, the Wess–Zumino–Witten equation takes the form $$$$\begin{equation*} \tilde{\tau }(f)=D_uD_uf+D_vD_vf+[D_uf,D_vf]=0, \end{equation*}$$$$where $$D_uf:=f_*(\partial _u)$$, $$D_vf:=f_*(\partial _v)$$, and for a vector field $$V$$ along $$f$$, $$D_uV$$ (resp., $$D_vV$$) is the covariant derivative of $$V$$ along $$f_*(\partial _u)$$ (resp., $$f_*(\partial _v)$$). In [25, ‘Lemma'], Goldstein shows that that $$W$$ lying in the kernel of $$dF|_f$$ is equivalent to the condition …”